A New Frequency Estimation Method for Equally and Unequally Spaced Data

Abstract

Spectral estimation is an important classical problem that has received considerable attention in the signal processing literature. In this contribution, we propose a novel method for estimating the parameters of sums of complex exponentials embedded in additive noise from regularly or irregularly spaced samples. The method relies on Kronecker's theorem for Hankel operators, which enables us to formulate the nonlinear least squares problem associated with the spectral estimation problem in terms of a rank constraint on an appropriate Hankel matrix. This matrix is generated by sequences approximating the underlying sum of complex exponentials. Unequally spaced sampling is accounted for through a proper choice of interpolation matrices. The resulting optimization problem is then cast in a form that is suitable for using the alternating direction method of multipliers (ADMM). The method can easily include either a nuclear norm or a finite rank constraint for limiting the number of complex exponentials. The usage of a finite rank constraint makes, in contrast to the nuclear norm constraint, the method heuristic in the sense that the problem is non-convex and convergence to a global minimum can not be guaranteed. However, we provide a large set of numerical experiments that indicate that usage of the finite rank constraint nevertheless makes the method converge to minima close to the global minimum for reasonably high signal to noise ratios, hence essentially yielding maximum-likelihood parameter estimates. Moreover, the method does not seem to be particularly sensitive to initialization and performs substantially better than standard subspace-based methods.

A survey of machine learning for big data processing

“…Instead, distributed frameworks with parallel computing are preferred. Alternating direction method of multipliers (ADMM) [72,73] serving as a promising computing framework to develop distributed, scalable, online convex optimization algorithms is well suited to accomplish parallel and distributed large-scale data processing. The key merits of ADMM is its ability to split or decouple multiple variables in optimization problems, which enables one to find a solution to a large-scale global optimization problem by coordinating solutions to smaller sub-problems.…”

A method for 3D direction of arrival estimation for general arrays using multiple frequencies

“…and the parameters {✓ k } k can be extracted from g as described in [13]. To clarify this statement, note that there are (many) rank K operators g whose symbols are exponential polynomials [16], i.e., in the representation (9), c k would be polynomials and the sum would contain fewer terms.…”

“…Following [13], we propose to solve (11) using ADMM, cf., e.g., [18]. The interpolation operators I j in (11) can be realized as the tensor-product of one-dimensional interpolation functions ' [20], [21].…”

“…The method relies on a low-rank matrix approximation formulation based on recent generalizations of Kronecker's theorem [12]- [14] and makes use of the following original key ingredients. First, in the case of a rectangular (but not necessarily equally spaced) grid X and one single wavelength , it is explained in [13,Section VI] how to retrieve the parameters ✓ k fromŝ (see also [15] for a related approach). The theoretical ground behind this approach (extension of Kronecker's theorem to several variables) is established in [16].…”

On the Structure of Positive Semi-Definite Finite Rank General Domain Hankel and Toeplitz Operators in Several Variables

“…The connection between low-rank Hankel and Toeplitz operators and matrices, and properties of the functions that generate them play a crucial role for instance in frequency estimation [7,32,[46][47][48], system identification [14,16,31,33] and approximation theory [4][5][6][8][9][10]42]. The reason for this is that there is a connection between the rank of such an operator and its generating function being a sum of exponential functions, where the number of terms is connected to the rank of the operator (Kronecker's theorem).…”